.TH std::acosh(std::complex) 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::acosh(std::complex) \- std::acosh(std::complex)

.SH Synopsis
   Defined in header <complex>
   template< class T >                       \fI(since C++11)\fP
   complex<T> acosh( const complex<T>& z );

   Computes complex arc hyperbolic cosine of a complex value z with branch cut at
   values less than 1 along the real axis.

.SH Parameters

   z - complex value

.SH Return value

   If no errors occur, the complex arc hyperbolic cosine of z is returned, in the range
   of a half-strip of nonnegative values along the real axis and in the interval [−iπ;
   +iπ] along the imaginary axis.

   Error handling and special values

   Errors are reported consistent with math_errhandling.

   If the implementation supports IEEE floating-point arithmetic,

     * std::acosh(std::conj(z)) == std::conj(std::acosh(z)).
     * If z is (±0,+0), the result is (+0,π/2).
     * If z is (x,+∞) (for any finite x), the result is (+∞,π/2).
     * If z is (x,NaN) (for any^[1] finite x), the result is (NaN,NaN) and FE_INVALID
       may be raised.
     * If z is (-∞,y) (for any positive finite y), the result is (+∞,π).
     * If z is (+∞,y) (for any positive finite y), the result is (+∞,+0).
     * If z is (-∞,+∞), the result is (+∞,3π/4).
     * If z is (±∞,NaN), the result is (+∞,NaN).
     * If z is (NaN,y) (for any finite y), the result is (NaN,NaN) and FE_INVALID may
       be raised.
     * If z is (NaN,+∞), the result is (+∞,NaN).
     * If z is (NaN,NaN), the result is (NaN,NaN).
    1. ↑ per C11 DR471, this holds for non-zero x only. If z is (0,NaN), the result
       should be (NaN,π/2).

.SH Notes

   Although the C++ standard names this function "complex arc hyperbolic cosine", the
   inverse functions of the hyperbolic functions are the area functions. Their argument
   is the area of a hyperbolic sector, not an arc. The correct name is "complex inverse
   hyperbolic cosine", and, less common, "complex area hyperbolic cosine".

   Inverse hyperbolic cosine is a multivalued function and requires a branch cut on the
   complex plane. The branch cut is conventionally placed at the line segment (-∞,+1)
   of the real axis.

   The mathematical definition of the principal value of the inverse hyperbolic cosine
   is acosh z = ln(z +
   √
   z+1
   √
   z-1).

   For any z, acosh(z) =

   √
   z-1
   √
   1-z

   acos(z), or simply i acos(z) in the upper half of the complex plane.

.SH Example


// Run this code

 #include <complex>
 #include <iostream>

 int main()
 {
     std::cout << std::fixed;
     std::complex<double> z1(0.5, 0);
     std::cout << "acosh" << z1 << " = " << std::acosh(z1) << '\\n';

     std::complex<double> z2(0.5, -0.0);
     std::cout << "acosh" << z2 << " (the other side of the cut) = "
               << std::acosh(z2) << '\\n';

     // in upper half-plane, acosh = i acos
     std::complex<double> z3(1, 1), i(0, 1);
     std::cout << "acosh" << z3 << " = " << std::acosh(z3) << '\\n'
               << "i*acos" << z3 << " = " << i*std::acos(z3) << '\\n';
 }

.SH Output:

 acosh(0.500000,0.000000) = (0.000000,-1.047198)
 acosh(0.500000,-0.000000) (the other side of the cut) = (0.000000,1.047198)
 acosh(1.000000,1.000000) = (1.061275,0.904557)
 i*acos(1.000000,1.000000) = (1.061275,0.904557)

.SH See also

   acos(std::complex)  computes arc cosine of a complex number
   \fI(C++11)\fP             (\\({\\small\\arccos{z}}\\)arccos(z))
                       \fI(function template)\fP
   asinh(std::complex) computes area hyperbolic sine of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{arsinh}{z}}\\)arsinh(z))
                       \fI(function template)\fP
   atanh(std::complex) computes area hyperbolic tangent of a complex number
   \fI(C++11)\fP             (\\({\\small\\operatorname{artanh}{z}}\\)artanh(z))
                       \fI(function template)\fP
                       computes hyperbolic cosine of a complex number
   cosh(std::complex)  (\\({\\small\\cosh{z}}\\)cosh(z))
                       \fI(function template)\fP
   acosh
   acoshf              computes the inverse hyperbolic cosine
   acoshl              (\\({\\small\\operatorname{arcosh}{x}}\\)arcosh(x))
   \fI(C++11)\fP             \fI(function)\fP
   \fI(C++11)\fP
   \fI(C++11)\fP
   C documentation for
   cacosh
